CONFINED ACOUSTIC PHONONS IN SI NANOMEMBRANES: IMPACT ON THERMAL

PROPERTIES

Emigdio Chávez Ángel

M.Sc. Physics

Tesi Doctoral Programa de Doctorat en Física

Thesis Director:

Supervisor:

Prof. Dr. Clivia M. Sotomayor Torres

Prof. Dr. Jordi Mompart

Departament de Física

Facultat de Ciències

Research supported by the Comisión Nacional Científica y Tecnológica de Chile (CONICYT)

2014

…“Gobernar es educar”… Pedro Aguirre Cerda,

Presidente de Chile, 1938-1944

…“La educación es un bien de consumo”… Sebastián Piñera Echenique,

Presidente de Chile 2010-2014

…“Vamos a invertir primero en educación, segundo en educación, tercero en educación. Un pueblo educado tiene las mejores opciones en la vida y es muy difícil que lo engañen los corruptos y

mentirosos”… José Mujica Cordano,

Presidente de Uruguay, 2010-2015

Antes de votar: lea, mire, y juzgue

A mi Madre

Acknowledgments

i

ACKNOWLEDGMENTS

The author ex presses h is gratitude t o t he C hilean g overnment f or a r esearch fellowship

through t he C omisión N acional C ientífica y T ecnológica d e C hile, C ONICYT, B ecas C hile

2010. A dditionally, t he a uthor acknowledge t he financial s upport f rom t he F P7 pr ojects

NANOFUNCTION ( grant nr . 257375) , N ANOPOWER ( grant nr . 256 959) and M ERGING

(grant nr. 309150), as well as the Spanish MINECO projects nanoTHERM (grant nr. CSD2010-

0044), ACPHIN (FIS2009-10150) and TAPHOR (MAT2012-31392).

First of all, I would like to express my deepest gratitude to my thesis director Prof. Dr Clivia

M. S otomayor T orres and a ll m y c olleagues i n the P hononic a nd P hotonic Nanostructures

group, P2N, a t t he Catalan Institute of Nanoscience and nanotechnology, ICN2, especially t o

Dr. F rancesc A lzina, D r Juan Sebastian R eparaz an d D r Jordi G omis B resco for their g reat

contribution of the development of this work. Also, I would like to thank to Dr John Cuffe from

Massachusetts I nstitute of T echnology, M IT, f or p roviding he lpful d iscussions during the

development of this work.

I would like to thanks to our collaborators from Technical Research Centre of Finland, VTT,

Dr Andrey Shchepetov, Dr Mika Prunnila, and Prof. Dr Jouni Ahopelto for providing silicon

membranes.

I w ould l ike t o t hanks Dr T homas D ekorsy a nd hi s g roup from U niversity of K onstanz,

Konstanz, Germany for the measurement of phonon lifetime accepting me for two stay abroad

in his group.

I would like to thank to Prof. Dr Bahram Djafari-Rouhani from the Institut d’Electronique,

de Microélectronique et de Nanotechnologie (IEMN), Lille, France, and Prof. Dr El Houssaine

El B oudouti f rom t he I nstitut d ’Electronique, de M icroélectronique et d e N anotechnologie

Acknowledgments

ii

(IEMN), L ille, F rance, and t he L DOM, F aculté d es Sciences, U niversité Mo hamed I, O ujda,

Morocco, f or t heir c ontributions t o the und erstanding of t he di spersion r elation i n phonon ic

crystals and the development of the PWE method.

I would l ike to thank to Prof. Dr. Gang Chen, Prof. Dr. Keith Nelson, Dr. John Cuffe and

Mr. Jeffrey Eliason from Massachusetts Institute of Technology (MIT) for the measurements of

thermal diffusivity using the Transient Thermal Gradient method.

I would l ike t o t hank all my f riends a nd colleagues f rom ICN2 especially t o Dr F rancesc

Alzina, Dr Claudia Simão, Dr Erwan Guillotel, MsC Noèlia Arias, Dr Jordi Gomis-Bresco, Dr.

Markus R aphael W agner, Dr B artlomiej G raczykowski, Dr Juan S ebastian R eparaz, Dr J uan

Sierra, Dr Nikolaos Kehagias and Ms Sweta Bhansali for their great contribution to my

scientific and social life. In particular, I would like to thank Jordi, whose ability to create new

devices, e quipment, experimental methods etc. is inspiriting. Especial acknowledge t o Noèlia

and Erwan for t heir indispensable work for t he s cientific development of group. El me u més

profund agraïment a F rancesc per proporcionar les fructíferes discussions científiques i també

per tota l'ajuda brindada durant el desenvolupament d'aquest treball.

I a m a lso g rateful to Clivia f or pr oviding m e t his g reat oppo rtunity t o w ork i n E urope.

Without her, nothing of the work developed in this thesis would have been possible.

Finalmente quisiera agradecer a mi familia, especialmente a mi madre, Delcira Ángel, quien

a pesar de todas las penurias pasadas nunca se can só de decirme “estudia”, la educación es e l

bien más preciado que te puedo dejar. Por eso y muchas cosas más siempre estaré agradecido de

mi vieja. Of course, my s incere thanks to Alexandra, who arrived in my darkest hours to put

meaning to silence and mute to my fatigue.

Y como diría el Gran Gustavo Cerati,

Gracias totales…

iii

iv

Abstract

v

ABSTRACT

The miniaturization trend of the technology has l ed to power level densities in excess 100

watts/cm2, which are in the order of the heat produced in a nuclear reactor. The need for new

cooling t echniques ha s p ositioned t he thermal management on t he s tage t he l ast y ears.

Moreover, t he e ngineering of t he t hermal c onduction ope ns a r oute to e nergy ha rvesting

through, for example, thermoelectric generation. As a consequence, control and engineering of

phonons in the nanoscale is essential for tuning desirable physical properties in a device in the

quest to find a suitable compromise between performance and power consumption.

In the present work we study theoretically and experimentally the thickness-dependence of

the thermal properties of silicon membranes with thicknesses ranging from 9 t o 2000 nm. We

investigate the dispersion relations and the corresponding modification of the phase velocities of

the a coustic modes us ing i nelastic B rillouin light scattering s pectroscopy. A r eduction o f t he

phase/group velocities of the fundamental flexural mode by more than one order of magnitude

compared to bulk values was observed and is theoretically explained. In addition, the lifetime of

the coherent a coustic phonon modes w ith frequencies up t o 500 GHz was a lso studied us ing

state-of-the-art u ltrafast p ump-probe: asynchronous opt ical s ampling ( ASOPS). W e ha ve

observed that the lifetime of the first-order dilatational mode decreases significantly from ∼ 4.7

ns to 5 ps with decreasing membrane thickness from ∼ 194 to 8 nm. Finally, the thermal

conductivity of membranes was investigated using three different contactless techniques known

as single-laser Raman t hermometry, t wo-laser R aman t hermometry an d t ransient t hermal

gradient. We have found that the thermal conductivity of the membranes gradually reduces with

their thickness, reaching values as low as 9 Wm-1K-1 for the thinnest membrane.

In order to account for the observed thermal behaviour of the silicon membranes we have

developed different theoretical approaches to explain the size dependence of thermal properties.

Abstract

vi

The simulation o f a coustic di spersion w as c arried out by us ing m odels based o n an el astic

continuum approach, Debye and fitting approaches. The s ize dependence of the lifetimes was

modelled considering intrinsic phonon-phonon processes and extrinsic phonon scatterings. The

thermal c onductivity w as modelled us ing a m odified 2D D ebye a pproach (Huang m odel),

Srivastava-Callaway-Debye model and Fuchs-Sondheimer approach.

Our o bservations h ave si gnificant co nsequences f or S i-based t echnology, e stablishing t he

foundation to investigate the thermal properties in others low-dimensional systems. In addition,

this study would provide design guidelines and enable new approaches for thermal management

at nanometric scales.

Table of contents

vii

TABLE OF CONTENTS

Acknowledgments ................................................................................................................................. i

Abstract .............................................................................................................................................. v

Table of contents ................................................................................................................................ vii

List of figures ................................................................................................................................... xiii

List of tables ................................................................................................................................... xxiii

List of publications and presentations ............................................................................................. xxv

Published and accepted articles................................................................................................. xxv

Non-related articles ................................................................................................................ xxvii

Book chapters ......................................................................................................................... xxvii

In preparation articles ............................................................................................................ xxviii

Oral Presentations.................................................................................................................. xxviii

List of acronyms .............................................................................................................................. xxxi

Chapter I: Introduction and Objectives ............................................................................................... 1

1.1 Nanoscale thermal conductivity .............................................................................................. 3

1.2 Phonon confinement................................................................................................................ 8

1.4 Thesis Outline........................................................................................................................ 10

Chapter II: Acoustic Waves ............................................................................................................... 11

2.1 Elastic continuum model ....................................................................................................... 11

2.1.1 Boundary conditions and confined waves ................................................................. 13

Table of contents

viii

Lamb waves ........................................................................................................................ 15

Symmetric and antisymmetric modes................................................................................. 16

Shear Waves ....................................................................................................................... 21

2.1.2 Layered systems ........................................................................................................ 22

From one layer to N layers ................................................................................................. 23

Example .............................................................................................................................. 24

Chapter III: Anharmonicity and Thermal Conductivity ................................................................... 27

3.1 Harmonic effect in crystals ................................................................................................... 27

3.2 Phonon-phonon interaction ................................................................................................... 28

3.2.1 Normal and Umklapp process ................................................................................... 31

Selection Rules: Normal and Umklapp processes. ............................................................. 34

3.3 Phonon lifetime: relaxation time approximation .................................................................. 35

3.3 Evaluation of phonon relaxation times ................................................................................. 37

3.3.1 Extrinsic relaxation times .......................................................................................... 38

Boundary scattering ............................................................................................................ 38

Impurity scattering.............................................................................................................. 40

3.3.2 Intrinsic relaxation times ........................................................................................... 41

Phonon density of states ..................................................................................................... 44

Einstein and Debye approximations ................................................................................... 45

3.4.3 Phonon-phonon interaction and the Debye approximation ....................................... 47

Numerical simulations ........................................................................................................ 52

Table of contents

ix

Bulk Umklapp-processes .................................................................................................... 52

Intrinsic sound absorption: Akhieser and Landau-Rumer mechanisms ............................. 57

3.4 Thermal conductivity: modelling and approximations ........................................................ 58

3.4.1 Specific heat capacity ................................................................................................ 60

3.4.2 Thermal conductivity in low dimensional systems ................................................... 61

Modified Debye-Callaway-Srivastava model: complete phonon-phonon scheme ............. 61

Fuchs-Sondheimer model: correction of the thermal conductivity expression .................. 63

Huang model: modified dispersion relation and correction of the thermal conductivity

expression ........................................................................................................................... 64

3.5 Phonon confinement and modification of specific heat capacity ........................................ 68

3.5.1 Modification of the specific heat capacity ................................................................. 68

3.6 Use of modified dispersion relation: defining criteria.......................................................... 72

Chapter IV: Fabrication and Experimental Techniques .................................................................... 74

4.1 Fabrication of ultrathin freestanding silicon membranes ..................................................... 74

4.2 Advanced methods of characterizing phonon dispersion, lifetimes and thermal

conductivity ................................................................................................................................. 76

4.2.1 Brillouin scattering .................................................................................................... 76

4.2.2 Pump-and-probe ultrafast spectroscopy. ................................................................... 82

Generation and detection of high-frequency phonons ........................................................ 82

Asynchronous Optical Sampling: ASOPS ......................................................................... 84

4.2.3 Raman Thermometry ................................................................................................. 86

Table of contents

x

Single Laser Raman Thermometry: 1LRT ......................................................................... 87

Measurement of the absorbed power .................................................................................. 91

Measurement of the temperature field by two-laser Raman thermometry: 2LRT ............. 92

4.2.4 Transient thermal grating (TTG) ............................................................................... 94

Chapter V: Modelling and Experimental Results ............................................................................. 99

5.1 Acoustic phonons dispersion relation in ultrathin silicon membranes ................................ 99

5.1.1 Flexural mode dispersion ........................................................................................ 100

5.2 Phonon lifetime: measurements and simulations. .............................................................. 102

5.2.1 Lifetimes of Confined Acoustic Phonons in Ultrathin Silicon Membranes ............ 103

5.2.2 Phonon lifetime: theoretical results ......................................................................... 105

5.3 Thermal conductivity: measurements and simulations ...................................................... 114

5.3.1 Reduction of the thermal conductivity in free-standing silicon nano-membranes

investigated by non-invasive Raman thermometry .......................................................... 116

5.3.2 A novel contactless technique for thermal field mapping and thermal conductivity

determination: Two-Laser Raman Thermometry ............................................................. 122

5.3.3 Transient thermal grating measurements: temperature dependence of thermal

diffusivity ......................................................................................................................... 129

5.4 Thermal rectification ........................................................................................................... 131

5.4.1 Modelling of thermal rectification in Si and Ge thin films ..................................... 132

Chapter VI: Conclusions and Future Work ..................................................................................... 137

6.1 Thesis Summary .................................................................................................................. 137

Table of contents

xi

6.2 Future work ......................................................................................................................... 140

Appendix I: Elastic continuum model ............................................................................................. 145

I.1 Strain .................................................................................................................................... 145

I.2 Stress .................................................................................................................................... 148

I.3 Hooke’s law ......................................................................................................................... 151

I.4 From strain-stress relation to equations of motion .............................................................. 153

I.5 Boundary conditions and Lamb waves ............................................................................... 154

Appendix II: ahnarmonicity and Thermal conductivity .................................................................. 157

II.1 Harmonic effect in crystals ................................................................................................. 157

II.2 Thermal conductivity models ............................................................................................. 159

II.2.1 Boltzmann equation ................................................................................................ 159

II.2.2 Kinetic theory ......................................................................................................... 161

II.2.3 Cattaneo equation: hyperbolic heat equation. ......................................................... 165

II.2.4 Callaway model ...................................................................................................... 166

II.2.5 Holland model......................................................................................................... 170

II.2.6 Holland-Callaway modifications ............................................................................ 172

II. 3 Boundary scattering processes .......................................................................................... 173

Appendix III: Modeling of thermal transport .................................................................................. 177

III.1 Calculation of thermal conductivity ................................................................................. 177

III.2 Modelling of thermal transport: 2LRT and FEM simulations ......................................... 180

Curriculum Vitae .............................................................................................................................. 187

Table of contents

xii

References ........................................................................................................................................ 189

List of figures

xiii

LIST OF FIGURES

Figure 2.1 Schematic representation of longitudinal and transversal waves ............................. 12

Figure 2.2 Left: Scheme of a f ree-standing m embrane. R ight: Sy mmetric and antisymmetric

waves. .......................................................................................................................................... 15

Figure 2.3 Decomposition of longitudinal and transverse wavevectors. ................................... 16

Figure 2.4 (a) Dimensionless acoustic dispersion relation, f·a, and (b) group velocity, vg, of Si

membrane for d ilatational ( red dot ted lines, DW), f lexural (black s olid lines, FW) and s hear

(blue dashed lines, SW) waves as a function of the dimensionless in-plane wavevector, q//·a. .. 19

Figure 2.5 Out-of-plane component of the wavevector, the red solid and blue dotted lines are ql

and qt wavevector component respectively for dilatational (a) and flexural (b) waves. ............. 20

Figure 2.6 Scheme of layered system .......................................................................................... 23

Figure 2.7 Scheme of symmetric three-layer system. Here “b” is the thickness of layer 1 and 3,

and “a” is the thickness of layer 2 of layered system ................................................................. 25

Figure 3.1 Diagrammatic representation of coordinates of a lattice point. ............................... 28

Figure 3.2 Diagrammatic representation of a phonon-phonon interaction. .............................. 32

Figure 3.3 Diagrammatic representation of N ormal and Umklapp p rocesses, left and right

respectively. ................................................................................................................................. 33

Figure 3.4 Construction for intersection of three phonons in a line for N process, adapted from

ref. [7]. ......................................................................................................................................... 34

Figure 3.5 Construction o f t he intersection of three phonons in a line t o i llustrate Umklapp-

process......................................................................................................................................... 35

List of figures

xiv

Figure 3.6 Wavelength-dependent specularity p (λ) as a function o f phonon w avelength λ for

roughness values of η = 0.5 nm (black), η = 1 nm (red), η = 2 nm (blue). ................................. 39

Figure 3.7 Schematic representation of three phonon-phonon scattering processes. ................ 41

Figure 3.8 (a-f): Areas of Integration in the x-x’ plane allowed for U-processes. Where xi = 1 to 6

are given in Table 3.2 .................................................................................................................. 50

Figure 3.9 Areas of Integration in the x-x’ plane allowed for N-processes. Where x i = 1 to 6 are

given in Table 3.2 ........................................................................................................................ 51

Figure 3.10 Relaxation rate, 1/τU, for bulk silicon at room temperature via class I (a) and class

II (b) event. .................................................................................................................................. 53

Figure 3.11 Relative contribution to the total intrinsic relaxation time for each processes and

event ............................................................................................................................................ 53

Figure 3.12 Total relaxation rate for Umklapp-processes. Grey dotted line denotes the different

zones where each processes dominate. ....................................................................................... 54

Figure 3.13 Relaxation rate as a function of temperature and reduced wavevector for different

phonon-phonon processes. .......................................................................................................... 55

Figure 3.14 Total relaxation rate f or class I event as a function of temperature and r educed

wavevector: (a) three-dimensional plot, (b) contour plot (isoline). ............................................ 56

Figure 3.15. Total relaxation rate for class II event as a function of temperature and reduced

wavevector: (a) three-dimensional plot, (b) contour plot (isoline). ............................................ 56

Figure 3.16 Modelling a nd c omparison o f t hermal c onductivity of free-standing s ilicon

nanowires ref. [45] ..................................................................................................................... 63

Figure 3.17 Number of discrete modes for 10 nm thick Si membrane a s a function of t he

dimensionless in-plane wavevector. ............................................................................................ 66

List of figures

xv

Figure 3.18 Specific heat of Si as a function of temperature for the bulk (blue dashed line) and

for 1 to 120 nm thick free standing membrane. ........................................................................... 69

Figure 3.19 (a) Specific heat capacity and temperature dependence of flexural (red line), shear

(blue line) and d ilatational (black line) polarizations for a 10 nm thick silicon membrane. For

comparison the dependence of the Si bulk is also plotted (green line). (b) Contributions of each

polarization to the t otal specific heat f or 10 nm and 1 nm thick silicon membrane. T he solid

(dotted) black, solid (dotted) red and solid (dotted) blue lines represent the polarization

contribution of f lexural, shear and di latational modes, respectively for 10 nm (1 nm) thick Si

membrane. ................................................................................................................................... 70

Figure 3.20 (a) N ormalized s pecific he at capacity as a f unction of temperature t he bl ue l ine

illustrates the bulk values. (b) Specific heat as a function of the membrane thickness at 300, 10,

4 and 1 K. .................................................................................................................................... 70

Figure 3.21 Spectral density of the heat capacity of a 10 nm thick Si membrane at 30 K (a) and

300 K (b) as a function of frequency. .......................................................................................... 71

Figure 3.22 Lattice thermal energy (red solid line) and spacing energy (black solid and grey

dashed lines) as a function of the temperature and thickness, respectively. (b) Magnified image

of the low temperature/thickness regime plotted in linear scale. ................................................ 73

Figure 4.1 (a) Typical SOI wafers used in the fabrication of the free-standing membranes with

a few 100s nm thick SOI film and BOX layer. (b) The SOI film is thinned by thermal oxidation.

The thermal process creates compressive stress in the f ilm, as shown by the arrows. (c) After

release t he m embrane i s r elaxed a nd t ends t o b uckle. ( d) O ptical m icrograph o f a rel eased

1.4x1.4 mm2 membrane with thickness of 9 nm. Courtesy of Prof. Dr. Jouni Ahopelto. ............ 75

Figure 4.2 Wavevector conservation in the photoelastic backscattering configuration. ........... 79

Figure 4.3 Wavevector conservation in backscattering configuration via the ripple effect. ...... 79

List of figures

xvi

Figure 4.4 (a) Schematics of apparatus used for backscattering configuration, (b) Photograph

of appa ratus us ed f or backscattering configuration. T PFI i s a Tandem F abry-Perot

Interferometer ............................................................................................................................. 80

Figure 4.5 Schematic of Tandem Fabry-Perot Interferometer, TPFI, manufactured by JRS .... 81

Figure 4.6 Typical Brillouin spectra recorded for 200 nm thick free-standing Si membrane. The

first two peaks nearest the central quasi-elastic peak are identified as the zero-order f lexural,

A0, and dilatational, S0, modes. The others belongs to f irst, and second order dilatational

modes, S1 and S2, respectively. Adapted from J. Cuffe and E. Chavez et at. [21,137] ............... 81

Figure 4.7 Schematic of the response of a semiconductor to an ultra-short pulse. Electrons are

excited from the valence band, VB, to the conduction band, CB, where they decay rapidly to the

bottom of the conduction band through electron-electron collisions and phonon emission. The

dynamics a re then d escribed b y a s lower d ecay involving el ectron-hole pair r ecombination,

carrier diffusion, and thermal diffusion. Courtesy of Dr. John Cuffe. ........................................ 84

Figure 4.8 Schematic pump-probe experiment: (a) Mechanical delay and ( b )ASOPS with two

mode-locked lasers, adapted from J. Cuffe [136] ....................................................................... 85

Figure 4.9 Schematic time de lay be tween pum p an d pr obe pul ses. F rom G igaoptics G mbH

website. ........................................................................................................................................ 86

Figure 4.10 Schematic e xamples of R aman s pectra as t hermometer: ( a) t ypical R aman

spectrum showing the anti-Stokes, Rayleigh, and St okes signal, (b) Redshift and br oadening of

the linewidth due to temperature increasing, adapted from [154]. ............................................. 88

Figure 4.11 Scheme of the Raman thermometry method. ........................................................... 90

Figure 4.12 Schematic c onfiguration f or t he incident, reflected and t ransmitted power

measurements. ............................................................................................................................. 92

List of figures

xvii

Figure 4.13 Schematic c onfiguration of the T wo-Laser R aman T hermometry T echnique

developed in this work. ................................................................................................................ 94

Figure 4.14 Typical time trace from a 400 nm thick Si membrane. The electronic response of

the sample is seen, which decays quickly to leave the thermal response. This decay can then be

fitted to extract the decay time, which is proportional to the thermal diffusivity. Courtesy of Dr.

J. Cuffe. ....................................................................................................................................... 95

Figure 4.15 Schematics of F our-beam T ransient T hermal G rating appa ratus adapt ed f rom

Johnson e t a l. [173]. The angle be tween the pump beams i s controlled by sp litting the beams

with a d iffraction grating (phase mask) with a w ell-defined pitch. T he pump beams are l ater

blocked, w hile the signal from t he pr obe beam that i s di ffracted from t he t hermal di ffraction

grating i s recorded. This signal is m ixed w ith an attenuated r eference b eam f or he terodyne

detection. ..................................................................................................................................... 97

Figure 5.1 Brillouin spectra as function of the angle of incidence, showing for the fundamental

flexural mode of a 17.5 nm Si membrane. ................................................................................. 101

Figure 5.2 Dispersion relation and phase velocity of the zero order flexural mode of a 17.5 nm

silicon membrane: experimental results (blue dots), simulation (black solid line) and quadratic

fit ( red das hed l ine). I nset ( a): Sc hematic r epresentation of t he di splacement f ields o f t he

flexural mode, courtesy of Dr. Jordi Gomis-Bresco. ................................................................ 102

Figure 5.3 Fractional c hange in r eflectivity as a function o f t ime in the 1 00 nm s ilicon

membrane. T he s harp i nitial c hange i s due to the e lectronic r esponse of the membrane. T he

subsequent w eaker o scillations a re d ue to the excited a coustic m odes. ( b) C lose-up of t he

acoustic modes after subtraction of the electronic response for membranes with 100 and 30 nm

thickness shown by the green and red line, respectively. The sinusoidal decay of the reflectivity

due to the first-order dilatational mode is clearly observed as a f unction of time, with a faster

List of figures

xviii

decay obs erved for t he thinner m embrane. T he time trace o f the 30 nm m embrane ha s b een

magnified by a factor of 10 for clarity. Adapted from J. Cuffe et al. [23]. ................................ 104

Figure 5.4 Experimental and theoretical phonon lifetime of the first-order dilatational mode in

free-standing silicon membranes as a function of frequency. Experimental data of free-standing

silicon m embranes w ith t hickness values ranging f rom appr oximately 194 to 8 nm ( red

dots) [23] and 222 nm (black dot ) [180]. Green l ine: extrinsic boundary scattering processes.

Blue line intrinsic three-phonon normal scattering processes. The total contribution, calculated

using M atthiessen’s r ule, i s s hown by t he s olid bl ack-dashed l ine. A dapted from J . C uffe et

al. [23]. ...................................................................................................................................... 105

Figure 5.5 Phonon-phonon processes in Si membranes: (a) Intrinsic relaxation rate as function

of the frequency. (b) Relative contribution to the total intrinsic lifetime of each phonon-phonon

processes as a function of the frequency. .................................................................................. 108

Figure 5.6 (a) T hermal c onductivity of S i m embranes nor malized t o the S i bul k v alue a s

function of the thickness. The experimental data were obtained from thermal transient gradient

(black dots), R aman thermometry ( red dots) and t wo-laser R aman thermometry ( green d ot)

methods, res pectively [184–186]. T he t heoretical d escription o f t he d ata u sing t he F uchs-

Sondheimer model is shown in blue solid l ine. (b) Theoretical lifetime of the thermal phonon,

τTH, as a function of thickness: black line includes modification of thermal phonon lifetime due

to the decrease of the thermal conductivity, blue and red lines: constant thermal phonon lifetime

of 17 and 6 ps are shown for comparison. ................................................................................ 110

Figure 5.7 Experimental and theoretical phonon lifetime of the first-order dilatational mode in

free-standing silicon m embranes as a f unction of frequency. D ata o f f ree-standing Si

membranes w ith t hickness ra nging f rom 194 t o 8 nm (green dot s) w ere t aken f rom

Reference [23] and t he da ta poi nt for a 222 nm thick m embrane ( violet do t) w as t aken from

List of figures

xix

reference [180]. Solid blue (a), red (b) and black lines (c) are the intrinsic Akhieser attenuation

dependence ca lculated f or t hermal ph onon lifetimes of 17 p s ( a) and 6 ps ( b), w hereas ( c)

includes the thickness-dependent. ............................................................................................. 112

Figure 5.8 (a) Frequency dependence of the Q-factor for different values of the lifetime of the

thermal phonon: bulk values, τTH,bulk = 3kbulk/(CVv2), 50% of the bulk value, τTH = 0.5τTH,bulk and

10% of t he bul k v alue, τTH = 0.1τTH,bulk. (b) E xperimental an d t heoretical qu ality f actor of

different phonon modes in a S i nano-resonator. T he experimental d ata ( red d ots) were taken

from t he Reference [187], bl ue-solid line s hows t he best fit. Inset SEM image of the nano-

resonator, courtesy of Dr. J. Gomis-Bresco. ............................................................................. 113

Figure 5.9 Typical thermal conductivity measurement diagram .............................................. 115

Figure 5.10 Theoretical and experimental absorptance, A, reflectance, R, and transmittance, T,

as a function o f m embrane t hickness. T he s olids l ines ar e c alculation ob tained f rom F abry-

Perot simulations, courtesy of Dr. Francesc Alzina. The solid dots are experimental data points.

Inset: diagrammatic Fabry-Perot effect in membranes ............................................................ 118

Figure 5.11 Calibration of the Raman shift of the LO Si mode as function of the temperature:

the red and green dots were extracted from the References [156,186], respectively. ............... 119

Figure 5.12 Raman s hift ( right ax is) of the l ongitudinal optical ( LO) Si p honon o f t he

membranes as a function of the absorbed power and membrane thickness. The left axis

represent the temperature obtained from the temperature dependence of the LO mode extracted

from the slope of the Figure 5.11. ............................................................................................. 119

Figure 5.13 Thermal conductivity of the membranes, κfilm/κbulk, normalized to the bulk Si value

as a function o f the t hickness ( solid re d d ots). A s re ference p revious w ork in SOI [2–4] and

membranes using TTG [184] are also shown. The theoretical description of the data using the

List of figures

xx

modification of the dispersion relation, Srivastava and Fuchs-Sondheimer models are shown in

green dotted, black dashed and blue solid lines, respectively. .................................................. 120

Figure 5.14 Three-dimensional contour plot of the thermal field distribution of a 250 nm thick

free-standing Si membrane. The isoline distribution of the thermal field is also shown in a lower

plane. The colour bar indicates the maximum temperatures reaches. ...................................... 123

Figure 5.15 Vertical and horizontal temperature cuts of the isoline thermal field distribution of

a 250 nm thick free-standing Si membrane. Note the high symmetric distribution in temperature

from both cuts. ........................................................................................................................... 124

Figure 5.16 Comparison of t he temperature m ap m easured and s imulated for a 1 µm th ick

silicon m embrane. T he solid l ines rep resent theoretical cu rves w ith d ifferent t hermal

conductivity values ranging from the bulk values (1) and decreasing progressively to 65% of the

bulk values (0.65). ..................................................................................................................... 125

Figure 5.17 Experimental temperature p rofile measured i n a 1 µm t hick S i membrane (green

and pur ple dot s). T he pur ple do ts c ome from of t he ne gative pa rt o f t he F igure 5.16 m irror

reflected to the right side. .......................................................................................................... 126

Figure 5.18 Comparison of t he temperature m ap m easured and s imulated for a 2 µm th ick

silicon membrane. The solid lines represent theoretical curves with thermal conductivity value

of 118 WK−1m−1. ........................................................................................................................ 127

Figure 5.19 Comparison of the measured and s imulated temperature map for a 250 nm thick

silicon m embrane. T he so lid r ed l ine rep resents t heoretical cu rves w ith thermal co nductivity

value of 81WK−1m−1. The bulk limit, black solid line, is shown by comparison. ....................... 127

Figure 5.20 Theoretical and experimental thermal conductivity o f the Si membranes,

normalized to the bulk Si value, κfilm/κbulk, as a function of thickness. The green solid dots show

List of figures

xxi

this work (2LRT) and as reference previous work data on SO I [2–4] and Si membranes using

TTG [184], and Single-Raman thermometry, 1LRT, are also shown. ....................................... 128

Figure 5.21 Thermal diffusivity as a function of grating spacing in a 200 nm thick Si membrane

at different temperatures. .......................................................................................................... 130

Figure 5.22 Temperature dependence o f t he t hermal di ffusivity for 100 a nd 20 0 nm t hick Si

membranes. The Si bulk values are also shown for comparison [191,192] .............................. 131

Figure 5.23 Two-segment sch emes for t hermal rectification: ( a) I n-plane Si -Si or Si -Ge

configuration. (b) Out-of- plane Si-Ge configuration. .............................................................. 133

Figure 5.24 Temperature d ependence o f t he i n-plane ( a) and out -of-plane ( b) t hermal

conductivity of Si and Ge thin films. All curves were calculated with a mass-defect scattering

parameter ( Γ) r eflecting t he na tural isotope c oncentration, w ith the e xception of the 500 nm

thick Si film for which an increased mass-defect scattering was introduced (10Γ). ................. 134

Figure 5.25 Calculated in-plane thermal rectification coefficient of (a) 500-200 nm Si-Si, (b)

800-200 nm G e-Si and (c) 80 -15 nm G e-Si s ystems. C alculated out-of-plane t hermal

rectification of a 100-20 nm S i-Ge system. For each configuration T H, was fixed and t he low

temperature was varied. ............................................................................................................ 135

Figure 6.1 SEM i mage o f t he or dered ( a) and di sorder ( b) phononi c c rystal f or t hermal

properties analysis. Courtesy of Dr. Marianna Sledzinska. ..................................................... 142

Figure I.1 Sectioned solid under external loading ................................................................... 148

Figure I.2 Stress components ................................................................................................... 148

Figure I.3 Traction on arbitrary orientation ............................................................................ 149

Figure I.4 Typical graph of non-linear equation of dilatational waves for a fixed dimensionless

wavevector, Q = 0.5, as a function of reduced frequency ......................................................... 156

List of figures

xxii

Figure II.1 Silicon dispersion relation, adapted from [212]. ................................................... 170

Figure II.2 Schematic ph onon s pectrum s howing z one di vision, 0.5b, and the extension i nto

second Brillouin zone. Adapted from [104]............................................................................... 172

Figure III.1 (a) P honon di spersion r elation of bulk s ilicon and ge rmanium sys tems:

experimental results (red and blue dots) from Ref. [176] and second-order polynomial f it (red

and blue solid l ine). (b) Red and blue l ines: calculated temperature-dependence o f the lattice

thermal c onductivity of b ulk s ilicon a nd ge rmanium, r espectively. R ed and blue do ts: t he

experimental data of silicon and germanium bulk respectively obtained from Ref. [70] .......... 180

Figure III.2 Normalized intensity of the monochromatic light with wavelength of 407 nm as a

function of travelled distance in bulk silicon. Three different stars at 250 (red), 1000 (blue) and

2000 ( green) nm ar e included to rough comparison w ith t he t hicknesses of t he s tudied

membranes. Inset: idem in double logarithmic scale for better visualization of the graph. ..... 182

Figure III.3 Absorptance ( A), r eflectance ( R), and t ransmittance ( T) as a function of the

thickness o f t he m embranes. So lid black, r ed and blue lines ar e t he c alculated A , R and T

coefficient c onsidering t he F abry-Perot e ffect. F or comparison t he bu lk l imit o f t he A and R

coefficients ar e s hown i n dashed g rey an d pi nk l ines, r espectively. N ote that the os cillating

behaviour is more appreciable at small thicknesses (a), while larger thicknesses the absorption

reaches quietly the bulk values (b). Courtesy of Dr Francesc Alzina. ...................................... 183

Figure III.4 Temperature d ependence o f t he b ulk s ilicon t hermal co nductivity. T he

experimental d ata ( red d ots) w ere o btained from R ef. [214]. T he s olid blue l ine i s a s econd-

order polynomial adjust. ........................................................................................................... 184

Figure III.5 (a) Geometry of the sample used in the simulations. (b) Representative simulated

temperature field. ...................................................................................................................... 185

List of tables

xxiii

LIST OF TABLES

Table 1.1 Thermal conductivity measurements in Si nanostructures ........................................... 7

Table 3.1 N and U-processes interactions. s is the polarization, L and T are the longitudinal

and transverse polarization, respectively. The processes described in the last row is only valid

for N-Processes ........................................................................................................................... 49

Table 3.2 Limit values for areas of integration in the x-x’ space, with SL and ST longitudinal and

transversal sound speed respectively. ......................................................................................... 50

Table 3.3 Silicon parameters used in the simulation of Umklapp-processes in the bulk system. 52

Table III.1 Silicon and Germanium parameters used in the calculations. .............................. 179

xxiv

List of publications & presentations

xxv

LIST OF PUBLICATIONS AND PRESENTATIONS

Published and accepted articles

1. J. Gomis-Bresco, D. Navarro-Urrios, M. Oudich, S. El-Jallal, A. Griol, D. Puerto, E.

Chavez, Y. Pennec, B. Djafari-Rouhani, F. Alzina, A. Martínez, C. M. Sotomayor Torres, “A

1D opt omechanical crystal w ith a c omplete pho nonic ba nd g ap”, Arxiv: 1401. 1691v2, 2014.

Accepted in Nat. Commun.

2. E. Chávez-Ángel, R .A. Zarate, J . G omis-Bresco, F . A lzina, C .M. S otomayor T orres,

“Modification of Akhieser mechanism in Si nanomembranes and thermal conductivity dependence

of the Q-factor of high frequency nanoresonators”, Accepted in Semicond. Sci. Tech.

3. J.S. R eparaz, E. Chávez-Ángel, M. R . W agner, J. G omis-Bresco, B . G raczykowski, F .

Alzina, and C . M. Sotomayor T orres, “Novel high resolution contactless t echnique for t hermal

field m apping and t hermal c onductivity de termination: T wo-Laser R aman T hermometry”, R ev.

Sci. Instr., Vol. 85, 034901, 2014.

4. V. A. Shah, S. D. Rhead, J. E. Halpin, O. Trushkevych, E. Chávez-Ángel, A. Shchepetov, V.

Kachkanov, N. R. Wilson, M Myronov, J. S. Reparaz, R. S. Edwards, M.R. Wagner, F. Alzina, I.

P. Dolbnya, D. H. Patchett, P. S. Allred, M.J. Prest, P.M. Gammon, M. Prunnila, T. E. Whall, E.

H. C . P arker, C .M. S otomayor T orres, D . R . L eadley, “ High q uality si ngle cr ystal G e n ano-

membranes for opto-electronic integrated circuitry”, J. Appl. Phys. Vol. 115, 144307, 2014.

5. E. Chávez-Ángel, J. S. Reparaz, J. Gomis-Bresco, M. R. Wagner, J. Cuffe, B. Graczykowski,

A. S hchepetov, H . J iang, M . P runnila, J. A hopelto, F. A lzina a nd C . M . S otomayor T orres,

“Reduction of the thermal conductivity in free-standing silicon nano-membranes investigated by

non-invasive Raman thermometry”, APL Mat., Vol. 2, 012113, 2014.

http://arxiv.org/ftp/arxiv/papers/1401/1401.1691.pdfhttp://arxiv.org/ftp/arxiv/papers/1401/1401.1691.pdfhttp://dx.doi.org/10.1063/1.4867166http://dx.doi.org/10.1063/1.4867166http://dx.doi.org/10.1063/1.4870807http://dx.doi.org/10.1063/1.4870807http://dx.doi.org/10.1063/1.4861796http://dx.doi.org/10.1063/1.4861796

List of publications & presentations

xxvi

6. J. C uffe, O . R istow, E. Chávez, A. S hchepetov, P -O. C hapuis, F . Alzina, M . H ettich, M .

Prunnila, J. Ahopelto, T. Dekorsy and C. M. Sotomayor Torres, “Lifetime of Confined Acoustic

Phonons in Ultra-Thin Silicon Membranes”, Phys. Rev Lett., Vol. 110, 095503, 2013.

7. E. Chávez, J. Gomis-Bresco, F. Alzina, J.S. Reparaz, V.A. Shah, M. Myronov, D.R. Leadley

and C .M. S otomayor T orres, “ Flexural m ode d ispersion i n ul tra-thin G e m embranes”, 14t h

International Conference on Ultimate Integration on Silicon (ULIS), Vol. 185, 2013.

8. E. Chávez-Ángel, F. Alzina, C.M. Sotomayor Torres, “Modelling of thermal rectification in Si

and Ge thin films”, ASME 2013 International Mechanical Engineering Congress and Exposition:

Heat Transfer and Thermal Engineering, Vol. 8C, V08CT09A013, 2013.

9. J. Cuffe, E. Chávez, A. Shchepetov, P.O. Chapuis, El Houssaine El Boudouti, F. Alzina, T .

Kehoe, J. Gomis-Bresco, D. Dudek, Y. Pennec, B. Djafari-Rouhani, M. Prunnila, J. Ahopelto, and

C.M. S otomayor T orres, “Phonons i n Sl ow Motion: D ispersion R elations in U ltra-Thin Si

Membranes”, Nano Lett., Vol. 12, 3569, 2012.

10. E. Chávez, J. Cuffe, F. Alzina and C.M. Sotomayor Torres, “Calculation of the specific heat

capacity in free-standing silicon membranes”, J. Phys.: Conf. Ser., Vol. 395, 012105, 2012.

11. J. Cuffe, E. Chávez, A. Shchepetov, P.O. Chapuis, El Houssaine El Boudouti, F. Alzina, T .

Kehoe, J. Gomis-Bresco, D. Dudek, Y. Pennec, B. Djafari-Rouhani, M. Prunnila, J. Ahopelto, and

C.M. Sotomayor Torres “Effect of the phonon confinement on the dispersion relation and the heat

capacity i n nanoscale Si m embranes”, ASME 201 2 I nternational M echanical E ngineering

Congress and Exposition: Micro- and Nano-Systems Engineering and Packaging, Vol. 9, 1081,

2012.

http://prl.aps.org/abstract/PRL/v110/i9/e095503http://prl.aps.org/abstract/PRL/v110/i9/e095503http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6523515http://dx.doi.org/10.1115/IMECE2013-63613http://dx.doi.org/10.1115/IMECE2013-63613http://dx.doi.org/10.1021/nl301204uhttp://dx.doi.org/10.1021/nl301204uhttp://iopscience.iop.org/1742-6596/395/1/012105/http://iopscience.iop.org/1742-6596/395/1/012105/http://proceedings.asmedigitalcollection.asme.org/proceeding.aspx?articleid=1751689http://proceedings.asmedigitalcollection.asme.org/proceeding.aspx?articleid=1751689

List of publications & presentations

xxvii

Non-related articles

12. S. F uentes, F . C espedes, P. M uñoz, E. Chávez and L. P adilla-Campos, “ Synthesis and

structural characterization of nanocrystalline BaTiO3 at various calcination temperature” J. Chil.

Chem. Soc. Vol 58, 2077, 2013.

13. S. Fuentes, E. Chávez, L . Padilla-Campos and D.E. Diaz-Droguett, “Influence of r eactant

type on the Sr incorporation grade and structural characteristics of Ba1−xSrxTiO3 (x=0−1) grown

by sol–gel-hydrothermal synthesis” Ceramics International, Vol. 39, 8823, 2013.

14. E. Chávez, S . F uentes, R . A . Zarate, L . P adilla-Campos, “ Structural anal ysis of

nanocrystalline BaTiO3”, Journal of Molecular Structure, Vol. 984, 131, 2010.

Book chapters

15. M. Mouis, E. Chávez-Ángel, F. Alzina, A. Shchepetov, J. Ahopelto, C. M. Sotomayor Torres,

A. Nassiopoulou, M. V. Costache, S. O. Valenzuela, B. Viala, D. Zakharov, Wensi Wang, Beyond

CMOS Nanodevices 1, Chapter 7: “Thermal Energy Harvesting”, John Wiley & sons, Inc., pp.

135-219, 2014.

16. D. Leadley, V. Shah, J. Ahopelto, F. Alzina, E. Chávez-Ángel, J. Muhonen, M. Myronov, A.

G. Nassiopoulou, H. Nguyen, E. Parker, J . Pekola, M. Prest, M. Prunnila, J. S. Reaparaz, C.M.

Sotomayor Torres, K. Valalaki and T. Whall, BEYOND-CMOS NANODEVICES 1, Chapter 12:

“Thermal isolator through nanostructuring”, John Wiley & sons, Inc., pp. 331-363, 2014.

http://dx.doi.org/10.4067/S0717-97072013000400038http://dx.doi.org/10.4067/S0717-97072013000400038http://dx.doi.org/10.1016/j.ceramint.2013.04.070http://dx.doi.org/10.1016/j.ceramint.2013.04.070http://dx.doi.org/10.1016/j.ceramint.2013.04.070http://dx.doi.org/10.1016/j.molstruc.2010.09.017http://dx.doi.org/10.1016/j.molstruc.2010.09.017http://onlinelibrary.wiley.com/doi/10.1002/9781118984772.ch7/summaryhttp://onlinelibrary.wiley.com/doi/10.1002/9781118984772.ch12/summaryhttp://onlinelibrary.wiley.com/doi/10.1002/9781118984772.ch12/summary

List of publications & presentations

xxviii

In preparation articles

17. F. A lzina, E. Chávez-Ángel, J.S. R eparaz, J. C uffe, J . G omis-Bresco, A . S hchepetov, J .

Ahopelto, a nd C . M . S otomayor T orres, I nvited r eview a rticle: “Silicon m embranes: a m odel

system to study heat transport”, to be published in APL Mat.

Oral Presentations

1. E. Chávez, F. Alzina and C. M. Sotomayor Torres, “Modelling of thermal rectification in Si

and Ge thin films” European Material Research Society 2014 spring meeting, E-MRS 2014, May

26-30, 2014, Lille, France.

2. E. Chávez-Ángel, F. Alzina and C. M. Sotomayor Torres, “Modelling of thermal rectification

in Si and Ge thin films” ASME 2013 International Mechanical Engineering Congress and

Exposition, ASME-IMECE 2103, Nov. 15-21, 2013, San Diego, USA.

3. E. Chávez, J .S. R eparaz, J . C uffe, J. G omis-Bresco, M . R . W agner, A . S hchepetov, M .

Prunnila, J. A hopelto, F . Alzina a nd C . M . S otomayor T orres, “ Thermal p roperties o f si licon

ultra-thin membranes: A theoretical and experimental approach” 2nd edition of largest European

Event in Nanoscience and nanotechnology, ImageNano 2013, April 23-26, 2013, Bilbao, Spain.

4. E. Chávez, J. Cuffe, F. Alzina, J. S. Reparaz, W. Khunsin, A. Goñi, C. M. Sotomayor Torres,

“Calculation of thermal properties in silicon nanostructures” XVIII Simposio Chileno de Física,

Nov. 21-23, 2012, La Serena, Chile.

List of publications & presentations

xxix

5. E. Chávez, J . C uffe, F . A lzina, C .M. S otomayor, “ Specific he at of ul tra-thin f ree standing

silicon membranes” 6th European Thermal Sciences Conference, Eurotherm 2012, September 04-

07, 2012, Poitiers, France.

6. E. Chávez, J. Cuffe, F. Alzina, C.M. Sotomayor, “Heat capacity on free standing membranes

included native oxide”, XIV International Conference on Phonon Scattering in Condensed Matter,

Phonons 2012, July 8-12, 2012, Ann Arbor, MI USA.

7. E. Chávez, P .O. Chapuis, J . C uffe, F . A lzina, C .M. S otomayor-Torres “ Relaxation r ate i n

ultra-thin s ilicon m embranes”, S ummer S chool: “Energy H arvesting at micro an d n anoscale”,

NiPS S ummer S chool 2011, August 1 -4, Wo rkshop “E nergy Man agement at Mi cro an d

Nanoscale”, August 4-5, Perugia, Italy.

8. E. Chávez, P .O. C hapuis, J. C uffe, F . A lzina, C .M. S otomayor-Torres “ Acoustic phonon

relaxation rates in nanometer-scale membranes”, Phononics 2011: First International Conference

on Phononic Crystals, Metamaterials and Optomechanics, Phononics 2011, May 29th to June 2nd

2011, Santa Fé, New Mexico, USA.

xxx

List of acronyms

xxxi

LIST OF ACRONYMS

1D One dimensional

2D Two dimensional

3D Three dimensional

1-LRT Single-laser Raman thermometry

2-LRT Two-laser Raman thermometry

a Thickness of the membrane

A Absorptance

ASOPS Asynchronous optical sampling

BHS Buffered hydrofluoric acid

BOX Buried oxide layer

BLS Brillouin light scattering

BTE Boltzmann transport equation

CCD Charge-couple device

CW Continuous wave

DOS Density of States

DSP Double side polished

DW Dilatational waves

FEM Finite element method

FDTR Frequency-domain thermoreflectance

FPI Fabry-Pérot interferometer

FW Flexural waves

FWHM Full-width-half-maximum

LA (L) Longitudinal acoustic

HF Hydrofluoric acid

ILS Inelastic light scattering

LO Longitudinal optic

MFP Mean free path

PBTE Phonon Boltzmann transport equation

R Reflectance

RIE Reactive ion etching

SAW Surface acoustic wave

List of acronyms

xxxii

SEM Scanning electronic microscopy

SOI Silicon-on-insulator

SW Shear waves

T Transmittance

TA (T) Transverse acoustic

TDTR Time-domain thermoreflectance

TMAH Tetramethyl ammonium hydroxide

TPFI Tandem Fabry-Pérot interferometer

TTG Transient thermal grating

Q-factor Quality factor

Chapter I

1

CHAPTER I: INTRODUCTION AND OBJECTIVES

Due t o t he l arge v ariety o f p romising t echnological ap plications, t he c oncept o f

nanotechnology has become one of the most important and exciting fields encompassing many

disciplines s uch a s phy sics, c hemistry, bi ology, medicine, engineering a mong ot hers. At

nanometric sca les, material pr operties c an be dramatically modified i n c omparison w ith t heir

bulk counterpart. This is, in part, due to the effects of quantum confinement and to the increase

of s urface-volume r atio. F or e xample, a s pherical particle w ith s ize of 30 nm ha s 5 % of i ts

atoms on its surface; at 10 nm this percentage has increased by 20% while at 5 nm atoms at the

surface acco unt f or almost 50% t he t otal number [1]. These f actors c an e ither en hance or

degrade elastic, reactive, thermal, optical and electrical characteristics among others.

Great i nnovations i n controlled micro and nanofabrication ha ve l ed t o t he realization of

novel materials, development of processes and unveiling of new phenomena at nanoscale which,

in tu rn, have spurred technology g rowth a t an astounding pa ce, w hile o ffering us t he first

building blocks for the next green-industrial revolution.

In this sense, the control of the charge and heat transport in low-dimensional semiconductor

structures has become a cornerstone in the development of this next technology revolution. This

is in part motivated by the increasing importance of thermal management as a c onsequence of

the large power d ensities r esulting from the continuous miniaturization of electronics

components. Moreover, the e ngineering of the t hermal conduction ope ns a route t o energy

harvesting t hrough, f or e xample, t hermoelectric generation. A s a consequence, c ontrol a nd

engineering of phonons in the nanoscale is essential for tuning desirable physical properties in a

device in the quest to find a suitable compromise between performance and power consumption.

Introduction & objectives

2

Developments arising from c onfinement of electronic charge and l ight i n nanostructures

have been widely researched in the context of information and communication technology. In

contrast, progress in the study of phonons as main actors in heat conductivity, carrier electronic

mobility, detection limits and emission time-scales, among others, has been slower in rate and

smaller v olume. An e xample of t he s till poo r state-of-the-art on th is to pic it that until now

essential p arameters r emains u nknowns su ch a s: the frequency-dependence o f t he Grüneisen

parameter, accurate measurements of phonon lifetime and/or phonon mean f ree path, the

influence of constant [2–5] or frequency-dependent surface roughness parameter [6,7], the limit

of diffusive/ballistic thermal transport and their associated temperature and thickness transition,

to name but a f ew. One example of t his l imited state-of-the-art is th at only last year an

experimental proof of the effective phonon mean free path in bulk silicon was obtained [8].

The l ack of the k nowledge of t hese pa rameters i s d ue, i n pa rt, to s ubstantial challenges

associated with their quantitative experimental determination and the corresponding theoretical

model. In t his s ense, o ne s et of s tructures t hat are attracting increasing attention for t hermal

studies is free-standing membranes. These include solids plates (slabs) or rods (bars) connected

to s olid s ubstrate b y th e e xtremities. F rom one-atom th ick layers, e. g. g raphene [9], t o h igh

purity and single-crystal structure, e.g. Si membranes [10], these structures have found use in a

wide v ariety of i nteresting a nd i mportant applications, i ncluding v ery s ensitive f orces [11],

mass [12] and p ressure sensors, low-loss m acromolecule se parators [13], bo lometers

platform [14,15] and optomechanical cavities [16] among others.

In addition, as there is no interference from a su bstrate and as they can be fabricated with

precise, controlled and reproducible fabrication processes, these nanoscale objects facilitate the

experimental analysis and comparison with theoretical models and are a text-book example of a

nanoscale sy stem. T heir physical p roperties, e .g. el ectrical an d t hermal p roperties, can be

Chapter I

3

dramatically different compared to a thick sample or bulk counterpart, by orders of magnitude.

All these characteristics are of special interest from experimental and theoretical point of view.

1.1 Nanoscale thermal conductivity

The understanding of h eat p ropagation and t hermal pr operties in low-dimensional

nanostructures ha s m otivated i ncreasing r esearch activity an d, u ndoubtedly, the thermal

conductivity, k, is one of the most important and fundamental physical quantities.

The t hermal co nductivity of a m aterial g overns its ability t o t ransport h eat a nd p lays a

fundamental role i n t he de sign a nd pe rformance of the t echnological de vices. The dominant

carrier o f t he h eat en ergy depends o n t he t ype o f m aterial. I t can b e t ransported v ia ch arge

carriers (electrons), l attice waves ( phonons), electromagnetic waves ( photons), o r sp in w aves

(magnons). For non-metal, semiconductor and alloy materials the dominant conduction carrier

is the lattice thermal conduction, i.e., by phonons. A phonon is a pseudo-particle which

represents quantized modes of the vibrational energy of an atom or group of atoms in a lattice.

Considering t hat phonons a re pseudo-particles, i t is p ossible t o associate to ea ch o f them an

energy ħω and a pseudo-momentum p = ħq, which obey Bose-Einstein statistics [17].

Similarly to the electron case, the phonon energy can be represented as a dispersion relation,

i.e., a relationship between the phonon frequency and its wavevector. The slope of a dispersion

relation c urves de termines t he phonon g roup v elocity, vg = dω/dq. F or t he bulk case, t he

dispersion r elation of pho nons with short wavevector can b e co nsidered l inear an d t he sl ope

represents the s ound v elocity in the m aterial. H owever, w hen w e d ecrease the characteristic

dimensions of the material, this linear dependence no longer holds, and many d iscrete modes

appear leading t o the quantization of the phonon energy. T his spatial confinement a ffects the

Introduction & objectives

4

phonon group velocity, density of s tates, specific heat capacity, e lectron-phonon and phonon-

phonon interactions, etc. [18–23]. Moreover the decrease in dimension sets an upper limit to the

phonon mean free path, because the acoustic wave cannot continue to travel in the media due to

the boundaries.

The recent e xperimental and t heoretical reports point t o a n enhancement of t he

thermoelectricity figure o f m erit, ZT = S2σT/k (where S is the S eebeck co efficient, σ the

electrical conductivity k the thermal conductivity and T is the temperature), in thin f ilms [24–

28], nanowires [29–33], superlattices [34–37] and suspended phononic crystals [38–40]. This is

primarily a r esult o f t he thermal c onductivity decrease compared t o t he bulk c ounterpart,

without a co rresponding d ecrease in e lectrical properties. The r eduction of t he t hermal

conductivity i n these sy stems h as b een a ssociated with two p rincipal f actors: ( i) t he

modification of the acoustic di spersion relation due to the additional periodicity ( superlattices

and phononic crystal structures) [41–43] or spatial confinement of the phonon modes (thin films

and nanowires) [44–47] and (ii) the shortening of the phonon mean free path due to the diffuse

scattering of phonons at the boundaries [2,48–50].

To model heat t ransfer in nanostructures, ad vanced t heoretical models are r equired w hich

correctly take into account the frequency dependence of phonon properties. The majority of the

models of the thermal conductivity are derived from the phonon Boltzmann transport equation

(PBTE) un der the s ingle m ode r elaxation-time a pproximation [17]. F or l ow-dimensional

systems Zou et al. [51] classified the theoretical models into three types. The first one takes the

bulk f ormulation f or the thermal conductivity, i ntroduces t he m odified di spersion relation

caused by the spatial confinement and adds a boundary scattering rate to the total scattering rate

through Mattiessen’s rule [44]. The second one uses the bulk dispersion relation and derives an

exact solution of the PBTE after introducing the diffusive boundaries conditions, according to a

Knudsen flow model [2,48,52]. The third model, proposed by Zou et al. [51], is a combination

Chapter I

5

of these two approaches. This model takes the modified expression of the thermal conductivity

including the Knudsen flow model in addition to the modified dispersion relation. More recently

Huang et al. [46] developed one- and two-dimensional expressions for the thermal conductivity

of nanowire and thin films, which include the modified expression of the relaxation time due to

the boundaries.

The experimental m easurement o f t he t hermal c onductivity i nvolves t wo st eps: t he

introduction o f thermal en ergy i nto t he sy stem (heating) and t he detection of the c hange o f

temperature or related physical properties due to the increase in thermal energy (sensing). Both

heating and sensing can be measured, mainly, by e lectrical, optical and/or the combination of

both methods. In Table 1.1 a summary of main measurements of Si nanostructures performed in

the last ten years is given.

Reference Type of Nanostructure

Relevant Dimensions

[nm]

Type of measurement

Temperature [K]

Thermal conductivity

at 300 K [W K-1 m-1]

Ma et al. [53]

(2013) Inverse opals 18-38 Electrical, 3ω 30-400 0.6-1.3

Grauby et al. [54]

(2013) Nanowires 50 & 200 Electrical, 3ω-SThM

Room temperature 22-150

Claudio et al. [55]

(2012)

Nanostructured Bulk 30-40

Electrical, commercial 2-300 15-24

Feser et al. [56]

(2012) Nanowires 110-150 Optical, TDTR Room temperature 12-40

Marconnet et al. [57]

(2012)

Periodic porous nanobridge 196 Electrical, 3ω

Room temperature 3.4-112

Introduction & objectives

6

Reference Type of Nanostructure

Relevant Dimensions

[nm]

Type of measurement

Temperature [K]

Thermal conductivity

at 300 K [W K-1 m-1]

Weisse et al. [58]

(2012)

Porous nanowires 300-350 Optical, TDTR

Room temperature 51-142

Kim et al. [59]

(2012)

Free-standing phononic crystal 500

Electrical, Joule heating

Room temperature 32.6

Fang et al. [60]

(2012)

Mesoporous nanocrystalline

thin films 140-340 Electrical, 3ω 25-315 0.23-0.32

Liu et al. [61]

(2011) Free-standing

membrane 500 & 700 Optical, Raman

thermometry Room

temperature 118 & 123

Wang et al. [62]

(2011) Nanocrystals 64-550 Electrical, 3ω 16-310 8-79

Yu et al. [38]

(2010)

Free-standing phononic nanomesh

22-25

Electrical, suspended heater & detector

80-320 1.5-17

Doerk et al. [63]

(2010) Nanowires 30-300 Optical, Raman thermometry

Room temperature 10-81

Tang et al. [39]

(2010) Holey Si 100

Electrical, suspended heater & detector

20-300 1.7-51

Schmotz et al. [64]

(2010)

Free-standing membrane 340

Optical,

thermal transient grating

Room temperature 136

Chapter I

7

Reference Type of Nanostructure

Relevant Dimensions

[nm]

Type of measurement

Temperature [K]

Thermal conductivity

at 300 K [W K-1 m-1]

Hochbaum et

at. [30] (2008) Nanowires 20-300

Electrical, suspended heater & detector

20-320 2.5-8.5

Boukai et al. [29]

(2008) Nanowires 10 & 20

Electrical, suspended heater & detector

100-300 0.76 & 5.7

Hao et al. [65]

(2006) Thin films 50 and 80

Electrical, suspended heater & detector

Room temperature 32 & 38

Ju [66]

(2005) Thin films 20-50

Electrical, on substrate heater

& detector

Room temperature 30-55

Liu et at. [4]

(2005) Free-standing membranes 30

Electrical, Joule heating 300-450 30

Liu et al. [49]

(2004) Free-Standing

membrane 20 & 100 Electrical, Joule

heating 20-300 22 & 60

Li et al. [45]

(2003) Nanowires 22-115

Electrical, suspended heater & detector

20-320 6.7-40.7

Table 1.1 Thermal conductivity measurements in Si nanostructures

The first thermal conductivity models for bulk systems [67–70] were based on the solution

of the phonon B oltzmann t ransport equation ( PBTE) unde r the s ingle m ode r elaxation time

approximation. This approach provides the simplest picture of phonon interactions considering

that each phonon mode has a single relaxation time independent of others modes, i.e., it assumes

that a ll ot her phono n ha ve t heir e quilibrium di stribution [17]. The c alculation o f t he t hermal

conductivity i n s emiconductor m aterial implies t he k nowledge of t hree m ajor frequency-

dependent parameters, i.e., specific heat, CV, phonon group velocity, vg, and the phonon mean

Introduction & objectives

8

free path, Λ. The expression for thermal conductivity from the kinetic theory of gases is given

by

Λ= gV vC31k [1.1]

Taking into consideration the contribution of each mode q with transverse (T) or longitudinal

(L) polarization s, Equation [1.1] becomes:

∑ +=qs

qsqsqsqsqsB

nnvTVK

)1(3

222

2

τωk [1.2]

where ħ is t he r educed P lack’s co nstant, V the total volume, T the t emperature, KB is t he

Boltzmann c onstant, ωqs the phonon f requency, vqs the g roup v elocity, nqs the B ose-Einstein

equilibrium phonon distribution function and τqs = Λqs/vqs the total relaxation time of each mode.

From E quation [1.2] it is c lear that to m odel the lattice th ermal c onductivity w e n eed the

dispersion relation, the t otal relaxation t ime of e ach m ode a nd a num erical s cheme f or

performing t he i ntegration w ithin t he B rillouin z one. T he phon on di spersion r elation c an be

calculated t hrough s everal m ethods. H owever, th e c alculation o f the intrinsic relaxation t ime

and the summation over the Brillouin zone can be very time-consuming and the knowledge of

the anharmonic phonon-phonon scattering strengths is not yet sufficiently-well established [71].

1.2 Phonon confinement

Acoustic phonons play an essential role in almost all the physical properties of a crystal. The

statistics of phonons and their interaction with others particles sets a limit to some properties,

such as: electrical and thermal conductivity, sound transmission, reflectivity of ionic c rystals,

inelastic s cattering o f light, s cattering o f X-Rays a nd ne utrons, l inewidth o f qua ntum dot

Chapter I

9

emission, m aximum pow er carried b y optical f ibers a nd s o on [72,73]. In t his sen se, t he

engineering of new devices able to generate and control phonons becomes an essential issue in

the development of future technologies.

The pioneering s tudies of confined waves were pe rformed by Lord Rayleigh [74] in 1885.

He d emonstrated, t heoretically, the existence of surfaces acoustics waves, S AW, which

propagates al ong t he p lane su rface o f an i sotropic solid half-space. T hese w aves ar e n on-

dispersive, with a velocity slightly smaller than of t he bulk shear waves and their amplitudes

decaying ex ponentially f rom t he su rface, i .e., these waves are co nfined in t he surface. Years

later, following the results of Lord Rayleigh others scientists developed this topic, particularly

Pochhammer, Love, Sezawa, S toneley, Lamb, among others. Following t he Pochhammer and

Rayleigh’s work Horace Lamb [75] described the characteristics of waves propagating in free-

standing plates.

Early experiments to detect optical phonons in confined systems were performed by Fasol in

1988 [76], who u sed R aman scat tering t echniques t o sh ow that t he w ave v ector o f o ptical

phonons of a ten monolayer thick AlAs/GaAs/AlAs superlattice are confined and can only take

values given by qz = nπ/Lz, where Lz is the thickness of the layers. This early experiment

demonstrated no t on ly t hat pho nons are confined i n na nostructures bu t a lso that t he

measurement of phonon wave vectors are well described by relatively simple continuum models

of phono n c onfinement. Concerning t o m embranes, t echnically an aco ustic cav ity, first

experimental observation of confined acoustic phon ons of n anometre-scale membranes was

reported in 1987 in suspended 20 nm thick Au films [77]. Other works in free-standing

membranes has been reported on 100 and 200 nm in SiN films in 2003 [78] and 30 nm Si films

in 2004 [79].

Introduction & objectives

10

1.4 Thesis Outline

In this thesis we report the study of thermal properties in free-standing silicon membranes. In

Chapter I I, an analytical model for wave propagation in i sotropic f ree-standing me mbranes is

developed. There the main concepts of the elastic continuum theory are described. Chapter III

theoretical models of the thermal properties are developed, including phonon lifetime, phonon

density of states, heat capacity and thermal conductivity. Chapter IV the fabrication of the

samples and the different characterization techniques used for the study the phonon-dependent

properties a re d escribed. T he ex perimental r esults an d t heir t heoretical d escription a re

developed in Chapter V . F inally, t he c onclusion, the summary of t he main results a nd f uture

extension of this work are shown in the Chapter VI.

Chapter II

11

CHAPTER II: ACOUSTIC WAVES

In this chapter, a semi-analytical model for wave propagation in free-standing membrane is

developed. T he f ree-standing m embrane consists of a solid pl ate, s lab, c onnected t o a solid

substrate by the extremities. The membrane system is treated as a semi-infinite system, i.e., with

infinite e xtensions in x and y directions b ut w ith a f inite extension in t he z component. The

acoustic waves sustained by this system are solutions of the elasticity equation of the material

with st ress-free conditions at t he bound aries at z = ±a/2, w here a is th e thickness o f t he

membrane. In addition, the acoustic wave propagation and the phonon dispersion relation in a

layered system are calculated.

2.1 Elastic continuum model

To calculate the dispersion relation the elastic continuum model is often used. In this model,

the discrete nature of the atomic lattice is ignored and the material treated as a continuum. This

model c an be de rived f rom t he t heory of l attice v ibrations by c onsidering t hat the lattice

deformations vary slowly on a scale determined by the range of the inter-atomic forces [80], and

is usually valid provided that the wavelength of elastic waves, λ, is significantly larger than the

atomic la ttice co nstant, a0, i .e., λ/a0 ≥ 20. This corresponds to wavelengths approximately

longer than 10 nm, or frequencies smaller than approximately 100 GHz [81].

Within this model, a displacement of the material causes a strain, which can be described in

terms of the strain tensor, S, and is related to the gradient of the displacement, �Ui/�xi. In the

presence of the strain, the material generates internal forces that return it to its original

Acoustic waves

12

positions, i.e., the equilibrium state. These forces are expressed in terms of the stress tensor, T,

which is related to the S tensor through the elastic constant tensor, C, as:

klijklij SCT = [2.1]

where Cijkl is a fourth order tensor, which has 81 components, but only 36 of these components

are independent, due to symmetry considerations [69,82–84]. In isotropic materials, waves can

travel equally well in all directions, and the elastic constant tensor can be further simplified to

have two independent components.

As shown in the Figure 2.1, depending on t he type of displacement, there are two types of

acoustic waves. First, longitudinal waves (LA) are such that the displacement is parallel to the

propagation di rection. S econd, t ransverse o r s hear w aves ( TA) ha ve t heir d isplacement i n a

plane parallel to the wavefront and consequently normal to the propagation direction.

Figure 2.1 Schematic representation of longitudinal and transversal waves

The elastic continuum model provides an adequate description of elastics waves and can be

used to d escribe c onfinement e ffects in na nostructures w hen the dimensions be come

comparable t o the w avelength. A full d escription of t he p ropagation of a coustic w aves

according to the continuum elasticity model is given in a number of comprehensive textbooks,

by a uthors s uch a s A uld [82], N ayfeh [69], S add [83] and R ose [84]. A d etailed a nalysis an d

development of the associated equations is described in Appendix I.

Chapter II

13

By describing strain S in terms of the displacement U and the stress T in terms of the strain

component Sij, the equation of motion can be written purely in terms of the displacement and the

C tensor as:

∂∂

+∂∂

∂∂

=∂

∂

k

l

l

kijkl

j

i

xU

xUC

xtU

21

2

2

ρ [2.2]

In isotropic materials, waves can travel equally well in all directions, and the elastic constant

tensor can be further simplified to have two independent components:

ijijkkij SST µdλ 2+= [2.3]

where λ is called Lamé’s constant and µ is referred to as the shear modulus. Then the equation

of motion is given by:

) ( )( 222222

UvvUvtU

TLT ∇⋅∇−+∇=∂∂

[2.4]

where U = (u, v, w) is the amplitude of the displacement vector, vL = [(l + 2µ)/ρ]1/2 and vT =

[µ/ρ]1/2 are the longitudinal and transversal velocities of acoustic waves in a g iven continuum

medium, respectively.

2.1.1 Boundary conditions and confined waves

The introduction of boundary conditions in infinite media changes the nature of the acoustic

propagation. SAW [74] are solution of the wave equation with st ress-free boundary condition

on the surface. These waves are non-dispersive and they have two components corresponding to

bulk shear and longitudinal waves, with a velocity lower than the bulk shear waves. The atomic

displacement forming those waves occurs in the sagittal plane, that is, the plane normal to the

Acoustic waves

14

propagation direction and their amplitude decays exponentially from the surface. The motion of

individual atoms is taken to be elliptical.

Following Rayleigh’s w ork o ther s cientist d eveloped t his t opic, p articularly Love a nd

Sezawa. Love found a purely shear waves, SW, with displacement normal to the sagittal plane

existing in a half-space covered with a layer of softer material [85]. Sezawa found that SAW,

with displacement in the sagittal plane, could exist in layered system [86]. Rayleigh, Love and

Sezawa waves all occur in surface-wave-based devices.

Following t he s ame c oncept Lamb described propagating waves i n i sotropic st ress-free

plates, i.e., plate waves [75]. The waves sustained by this type of structures are solutions of the

elasticity equation of material with stress-free conditions at the boundaries at z = ±a/2, where a

is the thickness of the plate, with infinite extent in x and y directions. For unsupported plates the

normal c omponents of the s tress t ensor v anish at the su rface. This sy stem h as t wo t ypes o f

solutions: solutions with displacements confined to the sagittal xz plane, which are called Lamb

waves and solutions w ith displacements p erpendicular t o the s agittal p lane ar e cal led s hear

waves, SW. Lamb waves can be further divided into two categories of modes. Those with out-

of-plane symmetric and antisymmetric displacements with respect to midplane of the plate are

known as dilatational waves, DW, and flexural waves, FW, respectively.

Chapter II

15

Figure 2.2 Left: Scheme of a free-standing membrane. Right: Symmetric and antisymmetric waves.

Lamb waves

The classical problem of Lamb’s wave propagation is associated with a wave motion i n a

stress-free and isotropic plate. The solution of this system can be uncoupled between two fields:

shear waves (0, v, 0) and sagittal waves (u, 0, w). The displacement fields of the sagittal waves

are f requently written using t he H elmholtz de composition [82]. This co nsists i n a l inear

combination of a scalar, φ, and vectorial,ψ, potentials functions:

j

kijk

ii xx

U∂∂

+∂∂

=ψεφ [2.5]

where εijk is the Levi-Civita te nsor and t he sc alar an d v ectorial p otential correspond t o

irrotational and r otational fields, r espectively. Introducing this solution in t he e quation of

motion, Equation [2.4], is p ossible to unc ouple t he po tentials generating two ha rmonic

equations:

2

2

22 1

tvL ∂∂

=∇φφ

[2.6]

2

2

22 1

tvi

Ti ∂

∂=∇

ψψ

[2.7]

which has the typically plane wave solutions:

)](exp[)()](exp[)(

//